Curved exponential families were defined by [1]} and the sufficient statistic with a random number of summands \(\sum _{k=1}^Dx_k\)  was derived by [2]}. Probability distribution (REF ) is a special case of the exponential family derived in [3]} [see their formula (2.6)]. A more general probability distribution of \(\mathbf {X}\)  is presented in Section .
